Rectangular function ($\mathrm{rect}(t)=1_{|t|<1/2}$)

The rectangular function has value one on $|t|<1/2$ and zero on $|t|>1/2$. Endpoint values are convention-dependent and do not affect its ordinary integrals or Fourier transform. Shifts, amplitude changes and time rescaling produce a rectangular pulse of arbitrary height, center and duration. With $\hat{f}(k)=\int f(t)e^{-ikt}dt$, for $a>0$ the width-$2a$ pulse $f(t)=1_{|t|<a}$ has Fourier transform $2\sin (ak)/k$, with value $2a$ at zero by a limit. Its convolution with itself is a rescaled triangular function, $(2a-|t|)$ on $|t|<2a$ and zero outside, because the convolution measures the overlap length of two intervals.